Let $X$ and $Y$ be independent standard Gaussian random variabless -- i.e., $N(0,1)$. Let $\sigma \in \mathbb{R}$. Define $$Z=X \cdot e^{\sigma \cdot Y}.$$ Is there a closed form expression for the density of $Z$?

The density can be expressed as either of the following two integrals. $$\mathsf{pdf}_Z(z) = \frac{1}{2\pi}\int_{-\infty}^\infty \exp\left(\frac{-z^2 \cdot e^{-2\sigma y} - y^2 - 2\sigma y}{2}\right) \mathrm{d}y$$ $$~~~~~~~~~~~~~~~~~~~=\frac{1}{2\pi|\sigma z|} \int_0^\infty \exp\left(\frac{-\sigma^2 x^2 - (\log(x/|z|))^2}{2\sigma^2}\right) \mathrm{d}x.$$

I think of $Z$ as being a symmetric version of the log-Normal distribution or, alternatively, a heavy-tailed version of the Normal distribution. I expect the density to resemble that of the log-Normal far from the origin, namely something like $e^{-(\log z)^2}$. I would like an exact expression in order to be able to compute things about this distribution. Alas, computing this is beyond my powers of integration. Any help would be appreciated.


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